The Stacks project

Definition 4.20.1. We say that a diagram $M : \mathcal{I} \to \mathcal{C}$ is codirected or cofiltered if the following conditions hold:

1. the category $\mathcal{I}$ has at least one object,

2. for every pair of objects $x, y$ of $\mathcal{I}$ there exist an object $z$ and morphisms $z \to x$, $z \to y$, and

3. for every pair of objects $x, y$ of $\mathcal{I}$ and every pair of morphisms $a, b : x \to y$ of $\mathcal{I}$ there exists a morphism $c : w \to x$ of $\mathcal{I}$ such that $M(a \circ c) = M(b \circ c)$ as morphisms in $\mathcal{C}$.

We say that an index category $\mathcal{I}$ is codirected, or cofiltered if $\text{id} : \mathcal{I} \to \mathcal{I}$ is cofiltered (in other words you erase the $M$ in part (3) above).